Role of elastin anisotropy in structural strain energy...

Biomech Model Mechanobiol (2011) 10:599–611DOI 10.1007/s10237-010-0259-x

ORIGINAL PAPER

Role of elastin anisotropy in structural strain energyfunctions of arterial tissue

R. Rezakhaniha · E. Fonck · C. Genoud ·N. Stergiopulos

Received: 8 October 2009 / Accepted: 20 September 2010 / Published online: 7 October 2010© Springer-Verlag 2010

Abstract The vascular wall exhibits nonlinear anisotropicmechanical properties. The identification of a strain energyfunction (SEF) is the preferred method to describe its com-plex nonlinear elastic properties. Earlier constituent-basedSEF models, where elastin is modeled as an isotropic mate-rial, failed in describing accurately the tissue responseto inflation–extension loading. We hypothesized that theseshortcomings are partly due to unaccounted anisotropic prop-erties of elastin. We performed inflation–extension tests oncommon carotid of rabbits before and after enzymatic degra-dation of elastin and applied constituent-based SEFs, withboth an isotropic and an anisotropic elastin part, on theexperimental data. We used transmission electron micros-copy (TEM) and serial block-face scanning electron micros-copy (SBFSEM) to provide direct structural evidence ofthe assumed anisotropy. In intact arteries, the SEF includ-ing anisotropic elastin with one family of fibers in thecircumferential direction fitted better the inflation–extensiondata than the isotropic SEF. This was supported by TEMand SBFSEM imaging, which showed interlamellar elastinfibers in the circumferential direction. In elastin-degradedarteries, both SEFs succeeded equally well in predicting

R. Rezakhaniha (B) · E. Fonck · N. StergiopulosLaboratory of Hemodynamics and Cardiovascular Technology,Ecole Polytehnique Federale de Lausanne, 1015,Lausanne, Switzerlande-mail: [email protected]

E. Foncke-mail: [email protected]

N. Stergiopulose-mail: [email protected]

C. GenoudFriedrich Miescher Institute for Biomedical Research,Basel, Switzerlande-mail: [email protected]

anisotropic wall behavior. In elastase-treated arteries fittedwith the anisotropic SEF for elastin, collagen engaged laterthan in intact arteries. We conclude that constituent-basedmodels with an anisotropic elastin part characterize moreaccurately the mechanical properties of the arterial wall whencompared to models with simply an isotropic elastin. Micro-structural imaging based on electron microscopy techniquesprovided evidence for elastin anisotropy. Finally, the modelsuggests a later and less abrupt collagen engagement afterelastase treatment.

Keywords Arterial wall biomechanics · Anisotropy ·Elastin · Ultrastructure · Rabbit carotid artery ·Constitutive equation

1 Introduction

Vascular tissue shows a nonlinear anisotropic behavior, asa result of material properties, structural arrangements andinterconnections of its main constituents i.e. elastin, collagenand vascular muscle cells (Silver et al. 2003; Vito and Dixon2003; Zhou and Fung 1997). Significant effort has been putinto developing constituent-based or structural strain energyfunctions (SEFs) for the vascular tissue, that take into accountthe individual mechanical contribution of each intramuralelement (Driessen et al. 2005; Gasser et al. 2006; Gundiahet al. 2009; Holzapfel et al. 2000; Zulliger et al. 2004a,b) andtherefore provide a solid base to study mechanical propertiesof the tissue in health and disease.

SEFs should provide a complete 3D description of thestress-strain field. For an axisymmetric vessel, this trans-lates into a good description (fit) of both the pressure-radius (P–ro) and the pressure-longitudinal force (P–Fz)curves, as those are measured in a typical inflation–extension

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experiment. Most of the previously published constituent-based SEFs have considered that collagen is the only wallelement contributing to the material anisotropy (Holzapfelet al. 2000; Zulliger et al. 2004a,b). Although these SEFs suc-cessfully describe P–ro curves of vessels, they fail to describesimultaneously both the P–ro and the P–Fz curves, in veins(Rezakhaniha and Stergiopulos 2008) as well as in arter-ies (Zulliger et al. 2004a,b). In a recent study on veins, wehave suggested an anisotropic SEF for elastin with fibersin the longitudinal direction, which, in conjunction with ananisotropic collagen description, significantly improved thequality of the P–ro and P–Fz curves (Rezakhaniha andStergiopulos 2008). Yet, as the anisotropy observed in themechanical behavior of elastin could be different in arter-ies and veins, the present study was designed to focus onthe existence and nature of elastin anisotropy in arteries.We hypothesized that a SEF with both elastin and collagenas anisotropic materials provides a better description of themechanical properties of the arterial wall compared to tradi-tional SEFs with anisotropic collagen and isotropic elastincomponents. Furthermore, we searched for direct micro-structural evidence to support our hypotheses by studyingthe structural organization of elastin within the arterial wall.Finally, we applied our suggested SEF, with an elastin aniso-tropic term, to both intact and elastase-treated arteries toobtain more information about the effects of elastin frag-mentation on structural links between collagen and elastin inthe arterial wall.

2 Methods

2.1 Experimental data

2.1.1 Biological specimens

Six common carotids were excised from young white NewZealand rabbits (3 ± 0.4 kg) from a local abattoir. The lengthof the arteries was measured before and after the excision todetermine the in vivo longitudinal stretch of the vessels. Thearteries were submerged in phosphate-buffered saline (PBS)and transported on ice to the laboratory. Immediately afterarrival, the arteries were cleaned from surrounding tissue.The common carotid arteries were typically 50–60 mm inlength, from which two 16-mm segments were excised. Onesegment was used as a control artery and the other for elastasetreatment.

2.1.2 Biochemical treatment

We followed the procedure described in detail in an earlierstudy performed in our laboratory (Fonck et al. 2007). In sum-mary, the artery segments were submerged in a PBS bath and

stretched to their in vivo longitudinal stretch. Elastase wasgradually added to the bath to reach a final concentration of6.85 U/ml. Each artery was left in the bath for 30 min at 37◦Cand washed afterward with fresh PBS. To inhibit the elastasetrapped in the arterial wall, the arteries were left for 30 minin a bath of 2 U/ml aprotinin (Elastin). The artery was thenwashed with PBS.

2.1.3 Geometric measurements

Before running inflation–extension experiments, rings of∼0.4 mm thick were cut from the end of the segments tomeasure the opening angle, wall thickness, and inner andouter diameter. Rings were placed in a Petri dish with PBSfor 20 min at 37◦C to reach an equilibrium state, and pho-tographs were taken under an upright microscope at ×20magnification (Axioplan2; Carl Zeiss Inc.). The rings weresubsequently cut open and placed in PBS for 30 min at 37◦Cto measure the opening angle.

2.1.4 Biomechanical testing

Arteries were tested by an inflation–extension devicedescribed in detail in our previous study (Rezakhaniha andStergiopulos 2008) following the protocol described byFonck et al. (2007). In summary, the arteries were stretchedto their in vivo length and then submitted to preconditioningcycles of inflation–extension (pressure range: 0–180 mmHg,approximate rate: 1.5 mmHg/s) to ensure repeatability ofthe curve. External diameter data was measured with aCCD micrometer (model LS-7030, Keyence). Pressure wasrecorded with a blood pressure transducer (BLPR, World Pre-cision Instruments) and the longitudinal force by a load cellforce transducer (FORT10, World Precision Instruments).The passive arterial state was achieved by adding sodiumnitroprusside to the bath (SNP, 10−4M).

2.1.5 Transmission electron microscopy

Two intact carotid arteries were removed from the animalpostmortem. These were then immersed in a solution of 2.5%glutaraldehyde and 2.0% paraformaldehyde in 0.1 M phos-phate buffer, pH 7.4 for a total time of 4 h at 4◦C. The arterieswere initially pressurized at 70 mmHg for ten minutes, thenpressure was released, and to ensure good fixation, the arter-ies were cut into small rings (width

Role of elastin anisotropy 601

resin was hardened for 24 h in a 65◦C oven. Semi-thin sec-tions, 1 micron thick, were cut from the trimmed blocks usingglass knives to check the orientation of the vessels, and qual-ity of the tissue, and then, thin 50-nm-thick sections werecut using a diamond knife (Diatome, Switzerland) and ultra-microtome (Leica UCT). These thin sections were furthercontrasted with lead citrate (Reynold’s stain), and imagescaptured digitally on a CCD camera (SIS Morada, Munich)inside a Philips CM10 transmission electron microscope ata filament voltage of 80 kV.

2.1.6 Serial block-face scanning electron microscopy(SBFSEM)

We used the same resin blocks prepared for the TEM.Image stacks were obtained using Serial block-face scanningelectron microscopy SBFSEM developed by Denk andHorstmann (Denk and Horstmann 2004). The equipmentis composed of a scanning electron microscope (SEM)(FEI Quanta 200 VP-FEG) combined with an ultramicro-tome (3View, GATAN Inc). The ultramicrotome repeat-edly removes uniform sections from the specimen surfaceand therefore exposes subsequent surfaces for block-faceimaging by the SEM. Because the tissue block position isfixed, image registration is inherent and image distortion isabsent, greatly simplifying volume reconstruction (Denk andHorstmann 2004). Images were done on three different sitesin the media at the accelerating voltage of 3.5 keV and in thelow vacuum mode (0.35 Torr). The section’s thickness was50 nm, and the obtained images were 2,048×2,048 pixelswith 8.9 nm per pixel.

3 Mathematical model

3.1 Background

The mathematical model presented in this work is anextension of our work on venous tissue (Rezakhaniha andStergiopulos 2008). We considered the artery as a thick-walled circular cylinder that undergoes inflation–extensionexperiments and shows a nonlinear, anisotropic and incom-pressible behavior. The material behavior is assumed to bepseudo-elastic i.e. loading and unloading curves can be rep-resented by separate elastic laws as proposed by Fung (Funget al. 1979). The principal stretch ratios are defined con-sidering the zero stress state of the arteries, as explained indetail in our previous work (Rezakhaniha and Stergiopulos2008). In the absence of vascular tone, we considered onlypassive properties of the arterial wall and therefore separatedour constituent-based strain energy function into two parts

representing the elastin and collagen components:

�passive = felast�elast + fcoll�coll (1)where felast and fcoll are the fractions of the wall cross-section area composed of elastin and collagen and �passive,�elast and �coll represent the SEF for the artery in its pas-sive state, the SEF for the network of elastin and the SEF forthe network of collagen, respectively. To be precise, felastis the fraction of the wall cross-section composed of elastinthat bears load, and therefore one should exclude the part ofelastin area that is fragmented and thus does not bear load.Since the arteries were taken from young healthy rabbits,we have assumed that the elastin fragmentation is still lowin intact arteries and that felast could be assumed to be thetotal wall cross-section composed of elastin. As for the elas-tase-treated arteries, we have assumed that elastin is almosttotally fragmented, and therefore felast has been taken equalto 0.0. In general, the sum of the fragmented elastin and thenonfragmented elastin is equal to the total wall cross-sectioncomposed of elastin, as assessed by histology.

To define the collagen SEF, we followed the sameapproach as Zulliger et al. and assumed collagen as a col-lection of wavy fibers arranged in two helical families offibers at angles α and −α to the circumferential direction. Adetailed description of the method can be found in the studyof Zulliger et al. (2004a,b). In short, the engagement of col-lagen fibers is assumed to take place in a statistical mannerand is represented by a log logistic distribution:

ρfiber(E) =

⎧⎪⎪⎨

⎪⎪⎩

0 for E ≤ 0kb

(Eb

)k−1

[

1+(

Eb

)k]2 for E > 0

(2)

E is the local strain in the direction of the fiber, b > 0 is ascaling parameter and k > 0 defines the shape of the distri-bution. The mode (peak value) of this statistical distributionis situated at b((k − 1)/(k + 1))1/k . Larger b results in alater engagement, and larger scale parameter k results in amore spread out distribution. However, one should be care-ful to compare the modeling parameters of the engagementpattern between two groups of arteries, if they have differentnatural zero stress states (ZSS). The reason is that, as E isdefined with respect to different zero stress states (ZSS), themodeling parameters will differ due to this fact alone. To beable to compare the engagement patterns, the PDFs shouldbe defined with respect to the same reference configuration.

The strain energy function of an individual collagen fiberis defined by:

�fiber(E) ={

0 for E ≤ 0ccoll

12 E

2 for E > 0(3)

where ccoll is an elastic constant associated with collagenfibers. The contribution of the ensemble of collagen fibers to

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the SEF is therefore given by:

�f

coll(E) = �fiber ∗ ρfiber

=∞∫

−∞ρfiber(E

∗)�fiber(E − E∗)d E∗ (4)

where � fcoll is the strain energy function for the ensemble ofthe fibers. Assuming that collagen fibers are at an angle αto the circumferential direction in the eθ –ez plane and theirdirection is represented by the unit vector vα , we can defineI4, an invariant of the Cauchy-Green tensor C with respectto vα , as :

I4 = vα · C · vα = λ2 = λ2θ cos2 α + λ2z sin2 α (5)

Similarly, we can define the invariant I′4 with respect to a

vector v′α having the angle −α with the circumferential direc-tion. As we assumed that fibers are organized symmetricallyaround the longitudinal direction, I

′4 and I4 are equal in this

case.Thus, the collagen SEF, with half of the fibers at angle

α and the other half at −α to the circumferential direction,becomes:

�coll = fcoll(

1

2�

fcoll

(I4 − 1

2

)

+ 12�

fcoll

(I ′4 − 1

2

))

(6)

3.1.1 Anisotropic strain energy function for elastin

To define the SEF for elastin, we propose a combinationof an orthotropic model, accounting for a subset of elastinfibers oriented in the circumferential direction and an isotro-pic model, accounting for the remaining elastin matrix:

�elast = �iso(I1) + �aniso(I ′′4 ) (7)where I1 is the first invariant of the Cauchy–Green deforma-tion tensor C and I ′′4 is an invariant of C with respect to theeθ defined as:

I ′′4 = eθ · C · eθ = λ2θ (8)eθ is the unit vector in the circumferential direction.

For the isotropic component of elastin, we used a neo-Hookean SEF:

�iso = cielast(I1 − 3) (9)where cielast represents the modulus for the isotropic elastincomponent. As for the anisotropic component, we assumedthe one-dimensional form of an incompressible neo-Hook-ean material for each fiber, undergoing a uniform cross-section deformation when stretched in the fiber direction.This results in the following formulation for the anisotropic

part of the SEF:

�aniso = caelast⎛

⎝I ′′4 +2

√I ′′4

− 3⎞

⎠ (10)

where caelast > 0 is an elastic constant enabling us to obtaina higher modulus for elastin in the circumferential direction.

Combining the individual SEF for collagen and elastin, weobtain the global SEF of the artery at its passive (no vasculartone) state as:

�passive = felast⎛

⎝cielast (I1 − 3) + caelast

⎛

⎝I ′′4 +2

√I ′′4

− 3⎞

⎠

⎞

⎠

+ fcoll(

1

2�coll

(I4 − 1

2

)

+ 12�coll

(I ′4 − 1

2

))

(11)

3.2 Comparing patterns of collagen engagement

The natural ZSS of the control arteries is different from theelastase-treated arteries. Therefore, the modeling parametersrelated to the Eq. 2 cannot be directly used to compare thepattern of collagen engagement. In order to compare collagenengagement characteristics, the probability density functionsof the control and the elastase-treated arteries should be pre-sented with respect to the same reference configuration.

If Ecand Et are green strains along the fibers, with respectto the ZSS of control and treated arteries, respectively, andρc (Ec) and ρt (Et ) are the corresponding PDFs,

∞∫

−∞ρc(Ec)d Ec =

∞∫

−∞ρt (Et )d Et = 1 (12)

If the function of Ec versus Et is monotonic, the above equa-tion results in,

ρc(Ec)d Ec = ρt (Et )d Et (13)To calculate the relation between Ec and Et , first, we haveassumed that a point on the wall of the control group is line-arly transformed to a point on the wall of the elastase-treatedarteries. For instance, a point on the midline of a controlartery would be transformed to a corresponding point on themidline of the treated one. Second, the changes in the ref-erence configuration, i.e the changes in the OA, the radiiand the length after elastase treatment have been taken intoaccount to calculate the relation between Ec and Et . Finally,we have assumed that the relation between Ec and Et for apoint on the midline could represent the mean transformationfor the whole arterial wall. In the range of applied deforma-tions, this assumption results in a maximum error of 14% incalculations of Et based on Ec.

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3.3 Fit function

Pressure-radius and pressure-longitudinal force curves werefitted to the experimental data by minimizing the followingfunction:

� = 12

1

m

m∑

i

(r modi − rexpi

σ ri

)2

+ 12

1

m

m∑

i

(F modi − Fexpi

σ Fi

)2

(14)

m is the number of experimental points measured at differ-ent pressures. Superscript mod denotes the values predictedby the mathematical model, whereas superscript exp thosemeasured experimentally. The index i denotes the pressureat which the corresponding outer radius, r , and the longi-tudinal force, F , were obtained. σ is the standard deviationof the experimental mean value of radius or force at a givenpressure and longitudinal stretch ratio. It is used as a weight-ing factor, giving more weight to the points with the leastvariation.

In order to check the fit quality, the weighted residual sumof squares for radius and force, W RSSr , W RSSF , have beencalculated based on the following formula:

W RSSk =m∑

i

(k modi − kexpi

σ ki

)2

k = r, F (15)

4 Results

P–ro and P–Fz curves from inflation–extension experimentson intact arteries and elastase-treated arteries are plotted inFig. 1. As seen from the P–ro curves, the vessel inflatesmore abruptly at lower pressures in elastase-treated arter-ies than the intact arteries and the final diameter reachedat 20 kPa is higher in elastase-treated arteries (1.76 mm)than intact arteries (1.46 mm). In addition, based on P–Fzcurves, the elastin degradation causes a decrease in the mea-sured longitudinal force in low pressures (between 0 and10 kPa) and at higher pressures, makes the curve less steep.Elastase treatment led to a 24, 9 and 27% increase in theexternal diameter, internal diameter and the axial lengthin the zero load configurations, respectively. The openingangle was substantially decreased from 107 ± 24 degreesin intact arteries to 11.5 ± 8.4 degrees in elastase-treatedarteries.

Figure 2 shows the P–ro and P–Fz curve fits based on theoriginal model, with an isotropic elastin term. Column (a)presents the fit from original Zulliger et al. model, when onlythe P–ro curve was considered to fit the data and no impor-tance was given to the P–Fz curve in the fitting process. In thiscase, the model closely fits the P–ro curve (W RSSr = 3.72),yet gave a relatively poor fit for the P–Fz curve (W RSSF =

0 5 10 15 200.5

1

1.5

2

2.5

Rad

ius

[mm

]

0 5 10 15 20

−40

−20

0

20

Pressure [kPa]

For

ce[m

N]

control arterieselastase−treated arteries

Fig. 1 Experimental data from control and elastase-treated arteries

643). When the P–ro and P–Fz curves were both consideredin the fitting process, i.e. function � was optimized, thisresulted in � = 1.189, and the fit is shown in column (b).The quality of the fit for the P–Fz curve was improved, andW RSSF was reduced to 55.26 at the expense of the qualityof the P–ro fit (W RSSr =23.23).

Figure 3 presents the fits of the modified model with ananisotropic elastin component applied on the experimentaldata of intact arteries. The fit was considerably improvedcompared to the one resulted from the original model withan isotropic elastin (WRSSr = 3.65, WRSSF = 9.04 for themodified model compared to WRSSr = 23.23, WRSSF =55.26 in the original model). The fit function � was reducedto 0.1909 compared to 1.189 from the best fit of the originalmodel (Fig. 1b).

The parameters used in minimizing � in the fit procedureare listed in Table 1 for the original model with isotropic elas-tin component and the modified model with an anisotropicelastin component having fibers in the θ -direction. The modelresulted in WRSSr = 157 and WRSSF = 8, 377 when thesame parameters as for the best fit of the modified modelfrom Table 1 were applied on the experimental data of elas-tase-treated arteries (Fig. 4a). To improve the quality of thefit, b and α were let free to change, while other parame-ters were held constant which resulted in b = 0.832 andα = 0.80 rad compared to the original values of b = 1.63and α = 0.68 rad. The result of the new fit is plotted in Fig. 3b(WRSSr = 7.85 and WRSSF = 32.23), and the parametersused are indicated in Table 2.

The probability density function of the log logistic distri-butions assumed for collagen engagement is plotted in Fig. 5.

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Fig. 2 Original model ofZulliger et al., with an isotropicterm for elastin, applied onintact arteries a) when only P–rois fitted and b) when both P–roand P–Fz are fitted

0 5 10 15 200.5

1

1.5

2

2.5

Rad

ius

[mm

]

0 5 10 15 20

−40

−20

0

20

Pressure [kPa]

For

ce[m

N]

0 5 10 15 200.5

1

1.5

2

2.5

Rad

ius

[mm

]

0 5 10 15 20

−40

−20

0

20

Pressure [kPa]

For

ce[m

N]

experimental datatheoretical data

(a) (b)

0 5 10 15 200.5

1

1.5

2

2.5

Rad

ius

[mm

]

0 5 10 15 20−50

−40

−30

−20

−10

0

10

20

30

Pressure [kPa]

For

ce [m

N]


Fig. 3 Modified model of Zulliger et al., with an anisotropic SEF forelastin, applied on intact arteries when both P–ro and P–Fz are fitted

Figure 5a and b shows the distribution for two sets of param-eters from Table 2, as assessed by the fitting of the model;first, k = 6.64 and b = 1.63 (parameters related to the best

Table 1 Parameters used in minimizing � in intact arteries

Strain energyfunction

Original model(with isotropicelastin)

Modified model(with anisotropicelastin)

Fitted parameters cielast = 17.99 KPa cielast = 18.39 KPacaelast = 13.59 KPa

k = 5.35 k = 6.64b = 1.96 b = 1.63α = 0.60 rad α = 0.68 rad

Experimentally felast = 0.3 felast = 0.3defined fcoll = 0.3 fcoll = 0.3parameters

Imposed on model ccoll = 200 MPa ccoll = 200 MPa� 1.189 0.1909

fit on the data of intact arteries) and second, k = 6.64 andb = 0.832 (related to the best fit of elastase-treated arteries).To compare the engagement pattern of collagen, Fig. 5c plotsthe resulted distributions from Fig. 5a and b with respectto the same reference configuration (elastase-treated group).The transformation has been obtained using Eq. 13 and con-sidering the changes in the geometry of the ZSS configurationafter elastase treatment. As seen from Fig. 5c, in elastase-treated arteries, collagen engages at higher values of straincompared to intact arteries. The collagen engagement is lessabrupt. The distribution is extended between Et = 0.2 and

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Fig. 4 Modified model ofZulliger et al., with ananisotropic term for elastin,applied on elastase-treatedarteries (a) with sameparameters as optimized for theintact arteries (based on Table 1)and (b) when parameters b andα are optimized

0 10 200.5

1

1.5

2

2.5

Rad

ius

[mm

]

0 10 20

−40

−20

0

20

Pressure [kPa]

For

ce [m

N]

0 10 200.5

1

1.5

2

2.5

Rad

ius

[mm

]

0 10 20

−40

−20

0

20

Pressure [kPa]F

orce

[mN

]


(b)(a)

2, while in intact arteries, collagen engages between Et = 0and 0.6.

Figure 6 shows a TEM image of the arterial cross-sec-tion in the media. Elastin, smooth muscle cells and col-lagen fibers are depicted in dark, medium and light gray,respectively. In the arterial cross-sections, interlamellar elas-tin fibers (arrows) made links between elastin lamina (aster-isks) and SMCs and were oriented in the similar directionas the SMCs. Figure 7 indicates a 2D SBFSEM imageof the arterial cross-section where the same pattern wasobserved but with the elastin in light gray and collagenin dark gray. Figure 8 illustrates an example of 3D SBF-SEM images. An intralamellar elastin feature is markedwith arrows in the image. In Fig. 9, one could follow thesame IEF structure in 3D by choosing proper planes ofviews. As seen from the planes A and B, the IEF runsobliquely in the θ -r plane making a slight angle (around2.5◦) with the lamellar elastin (i.e. circumferential direc-tion) and does not extend in the z-direction. In addition,3D SBFSEM images were examined in three different sites.IEFs appeared mainly in the circumferential direction in alllayers; however, they became less organized when locatedin layers closer to the intima compared to those closer

Table 2 Parameters used in Fig. 3 for elastase-treated arteries

Parameters fromthe best fit ofintact arteries

K and αoptimized forelastase-treatedarteries

Fixed parameters cielast = 18.39 KPa cielast = 18.39 KPacaelast = 13.59 KPa caelast = 13.59 KPaccoll = 200 MPa ccoll = 200 MPak = 6.64 k = 6.64b = 1.63α = 0.68 rad

Free parameters felast = 0.0 felast = 0.0fcoll = 0.3 fcoll = 0.3No optimization b = 0.832

α = 0.80 rad� 129.1 0.456

to the adventitia. Moreover, moving from the intima tothe adventitia, IEFs became more abundant and thicker indiameter.

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(a) (b)

(c)

Fig. 5 Probability density functions of the log logistic collagen distri-bution related to a the control group with respect to the control arteries’reference geometry (model parameters k = 6.64, b = 1.63, as fittedto the control group), b the elastase-treated group with respect to thetreated reference geometry (model parameters k = 6.64, b = 0.832, asfitted to the elastase-treated group and c the control group (solid line)and elastase-treated group (dashed line) with respect to the elastase-treated reference geometry

5 Discussion

Our results showed that including an anisotropic formulationfor the SEF of elastin, with one family of fibers in the circum-ferential direction, could significantly improve the quality ofP–ro and P–Fz curve fits (� value was reduced from 1.189 to0.1909). Assessment of arterial tissue ultrastructure, based onscanning microscopy techniques, revealed interlamellar elas-tin fibers running in the circumferential direction betweenthe elastin lamellar sheets, thereby providing substantial evi-dence for this anisotropy. These ultrastructural features wereformulated in an anisotropic strain energy function for elas-tin, which included an isotropic part, representing the elastinlamellar sheets, and an anisotropic part for the family of fibersin the circumferential direction. The results also indicatedthat in the absence of a structurally functional elastin, colla-gen engages less abruptly and the angle of collagen fibers isaltered.

Fig. 6 A TEM image of the arterial cross-section showing SMC’snucleus (N ) and cytoplasm (CY T ), elastin laminas (asterisks) andintralamellar elastin fibers (arrows)

Fig. 7 A 2D SBFSEM image of arterial cross-section showing SMC’snucleus (N ) and cytoplasm (CY T ), collagen fibers (C), elastin laminas(asterisks) and intralamellar elastin fibers (arrows)

5.1 Elastin in constituent-based SEFs

In the present study, we have suggested a transversely isotro-pic SEF for arterial elastin with elastin as a fiber-reinforcedmaterial having one family of fibers in the circumferentialdirection. Table 3 summarizes characteristics of some con-stituent-based SEFs previously reported in the literature. Thetable provides the properties of each mural constituent con-sidered for the formulation of the SEF and the type of theexperimental data used to validate these SEFs. In both theSEF of Holzapfel et al. (2000) and the model of Zulligeret al. (2004a,b), collagen is the only constituent that contrib-utes to the anisotropic properties of the vascular tissue, andelastin is considered as an isotropic material. These models

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Fig. 8 3D SFBSEM imagesshowing an arterial volume:IEFs (arrows) and Elastinlamella (asterisks) can be seenon the images. IEFs runobliquely, spread from elasticlamella and end in SMCs.Volume dimensions are18 µm, 10 µm and 5 µm

Fig. 9 An interlaminal elastinstructure as seen by 3DSFBSEM images. Theintersection of planes A and Bhave been chosen on an IEFstructure. The IEF does notextend in z direction and isalmost parallel with elastinlaminae thus in thecircumferential direction

were validated using P–ro curves of intact arteries. However,when forced to fit the P–ro and P–Fz curves simultaneously,they provided a poor fit (Rezakhaniha and Stergiopulos 2008;Zulliger et al. 2004a,b). Our results suggest that the short-comings of these models to fit the multidimensional data are,at least partly, due to the elastin definition in their constit-uent-based SEF. The collagen term in their SEF seems tobe appropriate, because it can successfully fit both the P–roand the P–Fz curves simultaneously in the absence of elastin(Fig. 4b). However, including other structural features couldalso play a role in improving the quality of the fit. Among

these features, one could mention the dispersion of the fibersor the multilayered structure of the arterial wall.

The present study indicates that an anisotropic formula-tion for elastin can improve the match between SEF andexperimental measurements. In a former study on isolatedarterial elastin (autoclaved and alkali treated), Gundiah andcolleagues tested pig thoracic aortas using a biaxial stretchersystem. They proposed an orthotropic material symmetry forarterial elastin, but suggested that an isotropic SEF couldsuccessfully describe the biaxial data (Gundiah et al. 2007,2009). Since the microstructure and mechanical behavior of

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Table 3 A summary of existing constituent-based SEFs

Experimentaldata used forvalidation

Elastin Collagen SMCs

Holzapfel et al. (2000) Various intact arteries Isotropic Anisotropic –

Two family of fibers

Orientation of fibers

Zulliger et al. (2004a,b) Various intact arteries Isotropic Anisotropic Contraction state



Waviness of fibers

Gundiah et al. (2007) Autoclaved andalkali-extractedporcine thoracicarteries

Isotropic (two orthogonalfamily of fibers with samematerial properties)

– –

Rezakhaniha and Stergiopulos (2008) Intact rabbit facial veins Anisotropic (one family offibers in the longitudinaldirection)

Anisotropic –



Waviness of fibers

normal arterial wall varies with location along the vasculartree and species (Humphery 2002), the observed differencein mechanical behavior of elastin could be species/locationdependent (pig thoracic aorta vs rabbit common carotid). Inaddition, the study of Gundiah et al. was based on techniquesof collagen digestion. Digestion techniques can affect themechanical properties of the resulting network. For exam-ple, hot alkali treatment causes fragmentation of the elastinnetwork (Daamen et al. 2001; Lillie and Gosline 1990). Finerfeatures, such as IEFs, might also be degraded, while largerfeatures remain. The damage in fine structures could affectthe degree of elastin anisotropy, but this needs to be verifiedthrough careful ultrastructural assessment with appropriateimaging techniques.

In a recent study on rabbit facial veins, we have suggestedan anisotropic SEF considering elastin as a fiber reinforcedmaterial with one family of fibers in the longitudinal direction(Rezakhaniha and Stergiopulos 2008). In the current studyon arteries, we observed likewise that an anisotropic formu-lation for elastin could improve the quality of fits. However,contrary to the longitudinal orientation of elastin fibers inveins, we had to consider the fibers in the circumferentialdirection to obtain the best fit in arteries. In the present study,the circumferential direction of fibers was justified by the cir-cumferential direction of IEF structures in arteries (Fig. 9). Inthe study on veins, no histological reason for fiber directionwas investigated and the results were only based on the qual-ity of the fits. Assuming that the direction of IEFs dictatesthe direction of fibers in the SEF, one can imagine axiallyoriented IEFs for the studied veins. A more detailed study

of elastin structure in veins and arteries could be helpful toclarify the subject.

The present model assumes that all elastin fibers are ori-ented in the circumferential direction. This assumption issupported by the visual inspection of microscopic images ofthe wall structure. Yet, realistically, the fibers are distributedaround some mean direction. Consequently, the dispersionof fiber orientation could influence the overall behavior ofthe model. Based on our results, the model with only onefamily of fibers in the circumferential direction could fit suc-cessfully the experimental data. Therefore, we believe thatconsidering all the fibers in an effective circumferential direc-tion is an acceptable assumption. A detailed future study onthe angular distribution of IEFs would be needed to validateour assumption and to define in greater detail the structuralproperties of the elastin fiber network.

As for the isotropic part of elastin, we have chosen aneo-Hookean SEF, suggested by Holzapfel and Weizsacker(Holzapfel and Weizsacker 1998). This was different fromour previous study on veins (Rezakhaniha and Stergiopulos2008) where the SEF suggested by Zulliger et al. was utilized(Zulliger et al. 2004a,b):

�iso = cielast(I1 − 3)3/2 (16)The model by Zulliger et al. was based on an improved fitto the gross mechanical response of arteries. Watton et al.recently compared the model of Zulliger et al. for elastin withthe neo-Hookean model and concluded that the neo-Hook-ean model captures more accurately the mechanical responseof elastin (Watton et al. 2009). In summary, they observed

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that the model suggested by Zulliger et al. does not yield agood fit to (uniaxial) experimental data within the physiolog-ical range of stretches. Furthermore, the infinitesimal shearmodulus is zero in the linear limit and therefore, violating abasic requirement of continuum mechanics. Finally, the SEFis not well defined for I1−3 < 0, i.e. if the material is slightlycompressible. Therefore, the Zulliger et al. SEF may give riseto numerical difficulties in finite element applications. If, onthe other hand, elastin is modeled as a rubber-like material,the neo-Hookean formulation has a sound theoretical basis(Treloar 1942).

5.2 Elastin 3D ultrastructure in arteries

By definition, constituent-based SEFs need to contain thestructural information of each constituent. In this study, wetried to explain the suggested form of the elastin anisot-ropy based on its 3D structure. The 3D structure of arte-rial elastin has been subject to some controversial studies.While in some species, including man, the individual fibersappear to coalesce to form continuous, uninterrupted sheets,other studies in the pig aorta indicate that the medial elas-tic fibers retain their individuality and do not form sheets(Clark and Glagov 1985; Grut et al. 1977). Most of theseprevious microstructural studies have used the informationfrom 2D views in which complex features can easily bemisinterpreted. Recently, Denk and Colleagues developed anew technique called the serial block-face scanning electronmicroscopy (SBFSEM) to produce nanostructural informa-tion in three dimensions (Briggman and Denk 2006; Denkand Horstmann 2004). O’Connell et al. used this novel tech-nique to obtain 3D volumetric information of aortic medialmicrostructure and described in detail the 3D organization ofeach wall element in rat abdominal aorta (O’Connell et al.2008). In the present study, we have used SBFSEM as well astransmission electron microscopy (TEM) to access the struc-tural organization of elastin in arteries. These techniques pro-vide a basis to define the form of the suggested anisotropicSEF for arterial elastin.

In our study, TEM and SBFSEM images suggesteda nearly circumferential direction of interlamellar elas-tin fibers. Using TEM, we observed elastin fibers goingfrom lamella to SMCs in a nearly circumferential directionbetween elastin sheets. These structures were not found intransversely cut sections and we could only detect a cross-sectional view of elastin fibers, suggesting that interlamellarelastin structures do not continue in the longitudinal planeand are limited to the circumferential–radial plane. This wasconfirmed by the 3D SBFSEM images (Fig. 9). These resultsconstitute the structural basis for the suggested anisotropicform of the arterial elastin, derived from the family of fibersin the circumferential directions.

Our results are in accordance with earlier ultrastructuralstudies of arteries (Clark and Glagov 1979, 1985; Dingemanset al. 2000; O’Connell et al. 2008). Specifically, in a recentstudy, O’Connell et al. described in detail the medial ultra-structure of rat abdominal aorta using SBF-SEM and confo-cal microscopy techniques. They reported three unique formsof elastin in the aorta i.e. lamellae, IEFs and radial elastinstruts. Of the total elastin volume, lamellae comprised 71%,IEFs 27% and radial elastin struts the remaining 2%. Simi-lar to the current study, O’Connell and colleagues reportedthat IEFs and SMC nuclei were preferentially oriented inthe circumferential direction, which is in agreement with ourfindings.

5.3 Collagen’s engagement pattern and anglein absence of elastin

To study interlinks between collagen and elastin structures,we have applied our suggested SEF on the experimental datafrom intact and elastin-degraded arteries. Experimental datafrom vascular tissues, where selectively one of the wall con-stituents is altered, could be useful in assessing the mate-rial behavior of each constituent, their functional couplingand interlinks between different wall elements. These dataare therefore helpful to develop the SEF of different wallconstituents. Among models with selectively altered constit-uents, elastase-treated arteries offer an attractive model toaccess the biomechanical properties of arteries after elastindegradation, as seen in pathologies such as aortic stiffen-ing with age and aneurysm formation (Dobrin 1989; Dobrinand Canfield 1984; Hayashi et al. 1987). Recently, Foncket al. studied the collagen engagement profile in intact andelastase-treated arteries by applying the SEF proposed byZulliger et al. on P–ro curves and suggested a clear interac-tion between collagen and elastin in the arterial wall (Foncket al. 2007). Following the same approach, in the presentstudy, we extended the work of Fonck et al., first by usingboth P–ro and P–Fz curves from intact and elastase-treatedarteries and second by applying the newly proposed SEFincluding the elastin anisotropic term.

The values reported for k and b by Fonck et al. are dif-ferent from the current study (Fonck et al. 2007). Foncket al. suggested peak values of collagen distributions atE = 0.912 in control arteries and 0.869 in elastase-treatedarteries compared to the corresponding 1.557 and 0.794 in thecurrent study. Fonck et al. assumed a circumferential direc-tion for collagen fibers and used only P–ro curves to fit thedata. On the contrary, we let free the angle of collagen fibers,which was necessary to fit simultaneously both P–ro and P–Fzcurves. The choice of angle of collagen fibers may have led todifferent values of k and b in these studies. More importantly,we used a neo-Hookean SEF (I1 − 3) as the isotropic termfor elastin. Fonck et al. applied a different SEF suggested by

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Zulliger et al. ((I1 − 3)3/2) which makes the elastin stiffer athigher strains and therefore alters the engagement parametersof collagen.

In addition, Fonck et al. tried to compare the engage-ment pattern of collagen fibers between control and elas-tase-treated groups and suggested a more abrupt andearlier engagement pattern for collagen after elastase treat-ment. This was achieved by comparing directly the fittedparameters and the corresponding distributions. However,since the natural reference configuration of the control andelastase-treated arteries is different, the comparison of dis-tributions is not straightforward. The PDF related to the con-trol group (Fig. 5a) is a function of the Green strain withrespect to the ZSS of the control group (Ec), while the PDFrelated to the treated groups (Fig. 5b) is a function of theGreen strain with respect the ZSS of the treated group (Et ).Therefore, if the PDFs are compared directly, the changesin the shape of the distribution could be merely because ofthe differences in the ZSSs. In order to compare the colla-gen engagement pattern, the PDFs should be represented asa function of the same Green strain (Fig. 5c). This can beachieved by using Eq. 13, once the relation between Et andEc is known.

To calculate the relation between Et and Ec, we haveassumed that the transformation for a point on the midlinecould represent the mean transformation law for the wholearterial wall. Let us consider a collagen fiber at the mid wallof a control artery, with an angle of 39◦ to the circumferentialdirection. After elastase treatment, this fiber is subjected toan elongation of 156% as a result of (a) the diameter expan-sion of 15%, (b) the axial elongation of 27% and (c) theOA decrease of 240% (from 107◦ to 31◦). This results inthe transformation of the PDF in Fig. 5a into Fig. 5c (solidline).

Figure 5c could be therefore used to elucidate the effect ofelastase treatment on the collagen engagement pattern. Theelastase treatment seems to result in a later and less abruptengagement of collagen fibers compared to the control group.This may suggest possible structural links between colla-gen and elastin which are altered after elastase treatment,resulting in a wavier structure and therefore a later and lessabrupt engagement pattern. Further studies are needed toelucidate the role of geometrical changes and possiblemechanical interactions between collagen and elastin follow-ing the elastin degradation.

As for the modification in the effective angle of collagenfibers (α), based on Table 2, the angle was increased by 18%after degradation of elastin. Only by letting free α, we couldfit both the P–ro and P–Fz curves simultaneously in bothintact and elastase-degraded arteries. We reported an angle of0.68 rad (39◦) for collagen fibers in intact arteries. Holzapfelet al. have chosen an angle of 29◦ in their original model(Holzapfel et al. 2000), Driessen et al. have taken α = 30◦ for

the media layer (Driessen et al. 2005). Zulliger et al. foundthis angle to be 35.0◦ in porcine carotid arteries (Zulligeret al. 2004a,b). Thus, the angle obtained in the present studyis close to previously reported values.

5.4 Limitations

The vascular smooth muscle (VSM) plays an importantrole in the mechanical properties of the arterial wall.We have neglected the contribution of VSM, because weintended to look only at the “passive” components of thewall, which are principally elastin and collagen and todo so we tested arteries under maximally dilated vascu-lar smooth muscle (VSM). This was achieved in our studyby adding a high dose of sodium Nitroprusside, whichaccording to earlier studies and literature guarantees max-imal vasodilation. Maximally dilated VSM is assumed tohave a negligible effect on mechanical properties of elas-tic arteries at its passive state. The assumption has beenon the basis that its passive elastic modulus is reported tobe about one order of magnitude less than that of elas-tin (VanDijk et al. 1984). We acknowledge the poten-tial shortcoming of our model, which derives from thefact that other wall components (proteoglycans, groundsubstance, etc.) were also assumed to have a negligiblemechanical effect when compared to the principal structuralcomponents, i.e. elastin and collagen. However, one couldassume that the neo-Hookean model not only accounts forthe isotropic part of elastin, but also includes other wallcomponents.

Electron microscopy images show that smooth musclecells are oriented in the same direction as Interlamellar elas-tin fibers (IEFs) and thus could contribute to the mechanicalproperties in the same direction as IEFs. Therefore, the factthat we attribute anisotropy of elastin solely to IEF may notbe entirely true and anisotropy at low strains could be alsothe result of the combined effect of the in-series ensembleof IEF and VSM. Earlier experimental (Roy et al. 2005) andtheoretical work (Roy et al. 2005) from our laboratory pointstoward this direction.

The model suggested in this study is a one-layer model,and structural properties have been homogenized throughoutthe arterial wall. Clearly, this does not reflect reality, know-ing that the adventitia is primarily comprised of collagen.However, the artery considered in this study (rabbit commoncarotid) is an elastic artery with the adventitia occupyingonly about 10% of the wall. Therefore, although a two-layermodel could more correctly capture the structural propertiesof the rabbit common carotid, the error of using a one-layermodel may be not so significant. Of course, even within themedia alone, we have a multilayer structure (lamellar units)and our model falls again short in that respect providing asimplified description of reality.

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6 Conclusions

We conclude that in addition to collagen, arterial elastin playsan important role in anisotropic behavior of the tissue. Therole of elastin, in anisotropic properties, is a result of its 3Dstructure, as observed by electron microscopy techniques,and affects the arterial mechanics mainly in the elastin-dominated region. Finally, elastin fragmentation appears toresult in a later and less abrupt engagement of collagen fibers.

Acknowledgments The authors would like to thank Dr. GrahamKnott and Stephanie Rosset at EPFL’s SV Electron microscopy facilitiesfor the assistance with specimen preparation and imaging, AlessandraGriffa and J. C. Sarria at EPFL’s BIOP facilities for their assistance inimage analysis and Dr. Bryn Martin for proofreading of the text. Thiswork was supported by the Swiss National Science Foundation (GrantNo. 325230-125445).

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http://dx.doi.org/10.1115/1111.2907749http://dx.doi.org/10.1115/1111.2907749

Role of elastin anisotropy in structural strain energy functions of arterial tissueAbstract1 Introduction2 Methods2.1 Experimental data2.1.1 Biological specimens2.1.2 Biochemical treatment2.1.3 Geometric measurements2.1.4 Biomechanical testing2.1.5 Transmission electron microscopy2.1.6 Serial block-face scanning electron microscopy (SBFSEM)

3 Mathematical model3.1 Background3.1.1 Anisotropic strain energy function for elastin

3.2 Comparing patterns of collagen engagement3.3 Fit function

4 Results5 Discussion5.1 Elastin in constituent-based SEFs5.2 Elastin 3D ultrastructure in arteries5.3 Collagen's engagement pattern and angle in absence of elastin5.4 Limitations

6 ConclusionsAcknowledgmentsReferences

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